Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#8062 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C2×C22⋊C4 |
| commutator | C2×C4 | C2×C42×C7 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| socle | C22×C7 | C2×#95 |
| prime radical | C24×C7 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C22×C42×C7 ◁ G | |
520 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,12,15,13,16,11,21)(10,17,20,22,18,23,19,14)(24,25)(26,33)(27,31)(28,37)(29,32)(30,35)(34,39)(36,38);(8,10)(9,14,16,22)(11,19)(12,20)(13,18)(15,17,21,23)(24,26,28,36,29,35,27,34)(25,30,31,38,32,33,37,39);(10,18)(14,22)(17,23)(19,20)(24,27,29,28)(25,31,32,37)(26,34,35,36)(30,38,33,39)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 1050);